Hindawi Publishing Corporation
Boundary Value Problems
Volume 2008, Article ID 742030, 11 pages
doi:10.1155/2008/742030
Research Article
Critical Point Theory Applied to a Class of the
Systems of the Superquadratic Wave Equations
Tacksun Jung1 and Q-Heung Choi2
1
2
Department of Mathematics, Kunsan National University, Kunsan 573-701, South Korea
Department of Mathematics Education, Inha University, Incheon 402-751, South Korea
Correspondence should be addressed to Q-Heung Choi, qheung@inha.ac.kr
Received 22 July 2008; Accepted 25 December 2008
Recommended by Martin Schechter
We show the existence of a nontrivial solution for a class of the systems of the superquadratic
nonlinear wave equations with Dirichlet boundary conditions and periodic conditions with a
superquadratic nonlinear terms at infinity which have continuous derivatives. We approach the
variational method and use the critical point theory which is the Linking Theorem for the strongly
indefinite corresponding functional.
Copyright q 2008 T. Jung and Q.-H. Choi. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we consider the existence of a nontrivial solution for the following class of
the systems of the superquadratic wave equations with Dirichlet boundary condition and
periodic condition
utt − uxx
vtt − vxx
π π
× R,
av Fu x, t, u, v, in − ,
2 2
π π
bu Fv x, t, u, v, in − ,
× R,
2 2
π
π
u ± , t v ± , t 0,
2
2
1.1
ux, t π ux, t u−x, t ux, −t,
vx, t π vx, t v−x, t vx, −t,
where F : −π/2, π/2 × R × R × R → R is a superquadratic function at infinity which
2
Boundary Value Problems
has continuous derivatives Fr x, t, r, s, Fs x, t, r, s with respect to r, s, for almost any x, t ∈
−π/2, π/2 × R. Moreover, we assume that F satisfies the following conditions:
F1 Fx, t, 0, 0 Fx x, t, 0, 0 Ft x, t, 0, 0 0; Fxx x, t, 0, 0 Ftt x, t, 0, 0 Fxt x, t, 0,
0 0, Fx, t, r, s > 0 if r, s / 0, 0, infx,t∈−π/2,π/2×R, |r|2 |s|2 R2 Fx, t, r, s > 0,
F2 |Fr x, t, r, s| |Fs x, t, r, s| ≤ c|r|ν |s|ν ∀x, t, r, s;
F3 rFr x, t, r, s sFs x, t, r, s ≥ μFx, t, r, s∀x, t, r, s;
F4 |Fr x, t, r, s| |Fs x, t, r, s| ≤ dFx, t, r, sδ1 Fx, t, r, sδ2 ;
where c > 0, d > 0, R > 0, μ > 2, ν > 1 and 1/2 < δ1 ≤ δ2 ≤ 1/r, for some 1 < r < 2.
As the physical model for these systems we can find crossing two beams with
travelling waves, which are suspended by cable under a load. The nonlinearity u models
the fact that cables resist expansion but do not resist compression. Choi and Jung investigate
in 1–3 the existence and multiplicity of solutions of the single nonlinear wave equation with
Dirichlet boundary condition.
Let us set
Lu, v Lu, Lv,
Lu utt − uxx .
1.2
Then, system 1.1 can be rewritten by
1
LU ∇ AU, U Fx, t, u, v ,
2
π
0
U ± ,t ,
0
2
1.3
Ux, t π Ux, t U−x, t Ux, −t,
where ∇ is the gradient operator, U vu , A b0 a0 ∈ M2×2 R.
√
√
We note that ab, − ab are two eigenvalues of the matrix A b0 a0 , and that
− abU2E ≤ AU, UR2 ≤ abU2E ,
U u, v.
1.4
Let λmn be the eigenvalues of the eigenvalue problem utt − uxx λu in −π/2, π/2 × R,
u±π/2, t 0, ux, t π ux, t u−x, t ux, −t.
Our main result is the following.
Theorem 1.1. Let F satisfy the conditions (F1), (F2), (F3), and (F4). Assume that
λ2mn − ab / 0 ∀m, n with m, n / 0, 0,
a > 0,
b > 0,
ab < 1.
Then, system 1.3 has a nontrivial solution u, v.
1.5
1.6
1.7
T. Jung and Q.-H. Choi
3
In Section 2, we obtain some results on the nonlinear term F. In Section 3, we approach
the variational method and recall the critical point theorem which is the linking theorem for
the strongly indefinite functional. This plays a crucial role to find a nontrivial solution. In
Section 4, we prove Theorem 1.1.
2. Some results on the nonlinear term F
The eigenvalue problem for ux, t,
utt − uxx λu
π
u ± , t 0,
2
in
−
π π
,
2 2
× R,
2.1
ux, t π ux, t u−x, t ux, −t
has infinitely many eigenvalues
λmn 2n 12 − 4m2
m, n 0, 1, 2, . . .,
2.2
and corresponding normalized eigenfunctions φmn m, n ≥ 0 given by
√
φ0n φmn
2
cos2n 1x
π
for n ≥ 0,
2.3
2
cos 2mt· cos2n 1x
π
for m > 0, n ≥ 0.
Let Q be the square −π/2, π/2 × −π/2, π/2 and H0 the Hilbert space defined by
H0 u ∈ L Q | u is even in x and t and
u0 .
2
2.4
Q
The set of functions {φmn } is an orthonormal basis in H0 . Let us denote an element u, in H0 ,
by
u
hmn φmn .
2.5
We define a subspace D of H0 as follows:
D
u∈
hmn φmn :
λ2mn h2mn < ∞ .
2.6
mn
Then, this space is a Banach space with norm
1/2
2
2
u λmn hmn
.
2.7
4
Boundary Value Problems
Let us set E D × D. We endow the Hilbert space E with the norm
u, v2E u2 v2 .
2.8
We are looking for the weak solutions of 1.3 in D × D, that is, u, v such that u ∈ D, v ∈ D,
Lu av Fu x, t, u, v, Lv bu Fv x, t, u, v. Since |λmn | ≥ 1 for all m, n, we have the
following lemma.
Lemma 2.1. i u ≥ uL2 Q , where uL2 Q denotes the L2 norm of u.
ii u 0 if and only if uL2 Q 0.
iii utt − uxx ∈ D implies u ∈ D.
Lemma 2.2. Suppose that c is not an eigenvalue of L : D → H0 , Lu utt − uxx , and let f ∈ H0 .
Then, one has L − c−1 f ∈ D.
Proof. Let λmn be an eigenvalue of L. We note that {λmn : |λmn | < |c|} is finite. Let
f
2.9
hmn φmn,
then
L − c−1 f 1
hmn φmn .
λmn − c
2.10
Hence, we have the inequality
L − c−1 f 2 λ2mn
1
2
λmn − c
h2mn ≤ C
h2mn
2.11
for some C, which means that
L − c−1 f ≤ C1 fL2 Q ,
C1 √
C.
2.12
By F1 and F3, we obtain the lower bound for Fx, t, u, v in the term of |u|μ |v|μ .
Lemma 2.3. Assume that F satisfies the conditions (F1) and (F3). Then, there exist a0 , b0 ∈ R with
a0 > 0 such that
Fx, t, r, s ≥ a0 |r|μ |s|μ − b0 ,
∀x, t, r, s.
2.13
Proof. Let r, s be such that r 2 s2 ≥ R2 . Let us set ϕξ Fx, t, ξr, ξs for ξ ≥ 1. Then,
ϕξ rFr x, t, ξr, ξs sFs x, t, ξr, ξs ≥
μ
ϕξ.
ξ
2.14
T. Jung and Q.-H. Choi
5
Multiplying by ξ−μ , we get
ξ−μ ϕξ ≥ 0,
2.15
hence ϕξ ≥ ϕ1ξμ for ξ ≥ 1. Thus, we have
Fx, t, r, s ≥ F x, t, √
Rr
Rs
√
r 2 s2
R
,√
r 2 s2
r 2 s2
μ
√ 2
r s2
≥ c0
≥ a0 |r|μ |s|μ − b0
R
μ
2.16
for some a0 , b0 , where c0 inf{Fx, t, r, s | x, t ∈ Q, r 2 s2 R2 }.
Lemma 2.4. Assume that F satisfies the conditions F1, F2, and F3. Then,
0, 0,
i Q Fx, t, 0, 0dx dt 0, Q Fx, t, u, vdx dt > 0 if u, v /
grad Q Fx, t, u, vdx dt ou, vE as u, v → 0, 0;
ii there exist a0 > 0, μ > 2 and b1 ∈ R such that
μ
Q
Fx, t, u, vdx dt ≥ a0 u, vLμ − b1
∀u, v ∈ E,
2.17
iii u, v → grad Q Fx, t, u, vdx dt is a compact map;
iv if Q uFu x, t, u, vvFv x, t, u, vdx dt−2 Q Fx, t, u, vdx dt 0, then grad Q Fx,
t, u, vdx dt 0;
v if un , vn E → ∞ and Q un Fu x, t, un , vn vn Fv x, t, un , vn dx dt − 2 Q Fx, t,
un , vn dx dt/u, vE → 0, then, there exists uhn , vhn n and w ∈ E such that
grad
Q
Fx, y, un , vn dx dt
uhn , vhn E
uhn , vhn 0, 0.
uhn , vhn E
−→ w,
2.18
Proof. i It follows from F1 and F2, since 1 < ν;
ii by Lemma 2.3, for U u, v ∈ E,
μ
Q
Fx, t, Udx dt ≥ a0 ULμ dx dt − b1 ,
where b1 ∈ R, thus, ii holds;
iii it is easily obtained with standard arguments;
iv it is implied by F3 and the fact that Fx, t, u, v > 0 for u, v /
0, 0;
2.19
6
Boundary Value Problems
v by Lemma 2.3 and F3, for U u, v,
uFu x, t, u, v vFv x, t, u, vdx dt − 2 Fx, t, u, vdx dt
Q
Q
μ
≥ μ − 2 Fx, t, u, vdx dt ≥ μ − 2a0 ULμ − b1 .
2.20
Q
By F2,
Fx, t, u, vdx dt ≤ C FU x, t, ULr ≤ C |U|ν Lr
grad
Q
2.21
E
for some 1 < r < 2 and suitable constants C ,C . To get the conclusion it suffices to estimate
μ
|U|ν /UE Lr in terms of ULμ /UE . If μ ≥ rν, then this is a consequence of Hölder
inequality. If μ < rν, by the standard interpolation arguments, it follows that |U|ν /UE Lr ≤
μ
CULμ /UE ν/μ
UlE , where l is such that l −1 ν/μ. Thus, we prove v.
Lemma 2.5. Assume that F satisfies the conditions F1, F2, F3, and F4. Then, there exist ϕ,
ψ : 0, ∞ → R continuous and such that ψs/s → 0 as s → 0, ϕs > 0 if s > 0,
i grad
ii
Q
Q
Fx, t, u, vdx dt2E ≤ ψ
Q
Fx, t, u, vdx dt, ∀u, v ∈ E,
uFu x, t, u, v vFv x, t, u, vdx dt − 2 Q Fx, t, u, vdx dt ≥ ϕu, v, ∀u, v ∈ E.
Proof. i By F4, for all U u, v ∈ E,
Fx, t, Udx dt ≤ FU x, t, ULr
grad
Q
E
≤ C1 Fx, t, Uδ1 Fx, t, Uδ2 Lr
≤ C2 Fx, t, Uδ1 Lr Fx, t, Uδ2 Lr
≤ C3 Fx, t, Uδ1 L1/δ1 Fx, t, Uδ2 L1/δ2
≤ C4 Fx, t, UδL11 Fx, t, UδL21
δ1
C5
Fx, t, Udx dt
Q
δ2 Fx, t, Udx dt
,
Q
2.22
where 1 < r < 1/δ1 , 1/δ2 < 2, C1 , C2 , C3 , C4 and C5 are constants. Since δ1 , δ2 > 1/2, we
prove i.
T. Jung and Q.-H. Choi
7
ii By F3,
uFu x, t, u, v vFv x, t, u, vdx dt − 2 Fx, t, u, vdx dt
Q
Q
Fx, t, Udx dt ≥ μ − 2
≥ μ − 2
Q
μ
a0 ULμ
2.23
− b1 .
Thus, we prove ii.
3. Variational approach and linking theorem
Now we are looking for the weak solutions of system 1.3. We shall approach the variational
method and recall the linking theorem for the strongly indefinite functional. We observe that
the weak solutions of 1.3 coincide with the critical points of the corresponding functional
I : E −→ R ∈ C1,1 ,
1
1
LU·Udx dt −
AU, UR2 dx dt −
Fx, t, u, vdx dt.
IU 2 Q
2 Q
Q
3.1
Now, we recall the linking theorem for strongly indefinite functional cf. 4.
Lemma 3.1 linking theorem. Let E be a real Hilbert space with E E1 ⊕ E2 and E2 E1⊥ . one
supposes that
I1 I ∈ C1 E, R, satisfies P.S.∗ condition;
I2 Iu 1/2Lu, u bu, where Lu L1 P1 u L2 P2 u and Li : Ei → Ei is bounded and
self-adjoint, i 1, 2;
I3 b is compact;
I4 there exists a subspace E ⊂ E and sets S ⊂ E, T ⊂ E and constants α > w such that:
i S ⊂ E1 and I|S ≥ α;
ii T is bounded and I|∂T ≤ w;
iii S and ∂T link.
Then, I possesses a critical value c ≥ α.
Let E− , E0 , E be the subspace of E on which the functional U → 1/2 Q LU·U is
positive definite, null, negative definite, and E− , E0 and E are mutually orthogonal. Let P be the projection for E onto E , P 0 the one from E onto E0 , and P − the one from E onto E− .
Let En n be a sequence of closed subspaces of E with the conditions
En En− ⊕ E0 ⊕ En ,
where En ⊂ E , En− ⊂ E− ∀n,
En and En− are subspaces of E, dim En < ∞, En ⊂ En1 , ∪n∈N En is dense in E.
Let PEn be the orthogonal projections from E onto En .
Let us prove that the functional I satisfies the linking geometry.
3.2
8
Boundary Value Problems
Lemma 3.2. Assume that the conditions 1.5, 1.6, and 1.7 hold. Then, for any F with (F1), (F2),
(F3), and (F4),
i there exist a small number ρ > 0 and a small ball Bρ ⊂ E with radius ρ such that if
U ∈ ∂Bρ , then
α inf IU > 0,
3.3
ii there is an e ∈ E , R > ρ and a large ball DR with radius R > 0 such that if
W DR ∩ E0 ⊕ E− ⊕ {re | 0 < r < R},
3.4
sup IU ≤ 0.
3.5
then
U∈∂W
Proof. i By 1.7 and i of Lemma 2.4, we can find a small number ρ such that, for U ∈ E ,
1
LU·U −
AU, UR2 −
Fx, t, u, vdx dt
2 Q
Q
Q
√ ab
1
U2E − 0 UE .
≥
1−
2
λ00
1
IU 2
3.6
√
Since ab < 1 λ00 , there exist a small number ρ > 0 and a small ball Bρ ⊂ E with radius ρ
such that if U ∈ ∂Bρ , then inf IU > 0. Thus, the assertion 1 holds;
ii let us choose an element e ∈ E . Let U / 0, 0 ∈ E0 ⊕ E− ⊕ {re | r > 0}. We note that
if U ∈ E , then LU·U − AU, UR2 dx dt ≥ τ1 U2E ,
Q
−
if U ∈ E , then
Q
3.7
LU·U − AU, UR2 dx dt ≤
−τ2 U2E
for some τ1 > 0, τ2 > 0. Let us choose a sequence Un n , Un un , vn /
0, 0 ∈ E0 ⊕ E− ⊕ {re |
r > 0} such that Un E → ∞. Let us set Ŭ n Un /Un E . By Lemma 2.3, we have that
IUn b0
μ
μ−2
≤ L − AP Ŭ n 2E − a0 Ŭ n Lμ Un E − τ2 P − Ŭ n 2E
2
Un 2
Un E
L − A
r 2 e2E
Un 2E
−
μ
μ−2
a0 Ŭ n Lμ Un E
b0
− τ2 P − Ŭ n 2E .
Un 2
3.8
T. Jung and Q.-H. Choi
9
Since Un E → ∞, two possible cases arise. For the case Ŭ n Lμ → 0 it follows that Ŭ n 0,
hence P Ŭ n → 0 and P 0 Ŭ n → 0. Thus P − Ŭ n E → 1. Hence
Iun ≤ −τ2 .
2
U
n→∞
n E
3.9
lim sup
For the case Ŭ n Lμ ≥ ε > 0 3.6 implies
Iun lim
n → ∞ Un 2
E
−∞.
3.10
Thus, we can choose a large number R > 0 and a large ball DR ⊂ E0 ⊕ E− with radius R > 0
such that if W DR ∩ E0 ⊕ E− ⊕ {re | 0 < r < R}, then supU∈∂W IU ≤ 0. So the assertion
ii holds.
We shall prove that the functional I satisfies the P.S.∗c condition with respect to En n
for any c ∈ R.
Lemma 3.3. Assume that the conditions 1.5, 1.6, and 1.7 hold. Then, for any F with (F1), (F2),
(F3), and (F4), the functional I satisfies the P.S.∗c condition with respect to En n for any real number
c.
Proof. Let c ∈ R and hn be a sequence in N such that hn → ∞,Un n be a sequence such
that
Un un , vn ∈ Ehn , ∀n, IUn −→ c, PEhn ∇IUn −→ 0.
3.11
We claim that Un n is bounded. By contradiction we suppose that Un E → ∞ and set
n Un /Un E . Then
U
n ∇IUn , U
n
PEhn ∇IUn , U
3.12
∇Fx, t, Un ·Un dx dt − 2 Q Fx, t, Un dx dt
IUn Q
−
−→ 0,
2
Un E
Un E
hence
Q
∇Fx, t, Un ·Un dx dt − 2 Q Fx, t, Un dx dt
Un E
−→ 0.
3.13
By v of Lemma 2.4,
grad
Q
Fx, t, Un dx dt
Un E
converges
3.14
10
Boundary Value Problems
n 0. We get
and U
PEhn grad Q Fx, t, Un dx dt
PEhn ∇IUn n − AU
n −
PEhn LU
−→ 0,
Un E
Un E
3.15
n − AU
n converges. Since U
n n is bounded and L − A is a compact mapping,
so PEhn LU
n
n 0, 0, we get U
n → 0, 0, which is a
up to subsequence, Un n has a limit. Since U
n E 1. Thus Un n is bounded. We can now suppose that
contradiction to the fact that U
Un U for some U ∈ E. Since the mapping U → grad Q Fx, t, Udx dt is a compact
mapping, grad Q Fx, t, Un dx dt → grad Q Fx, t, u, vdx dt. Thus, PEhn LUn − AUn n
converges. Since L − A is a compact operator and Un n is bounded, we deduce that, up to a
subsequence, Un n converges to some U strongly with ∇IU lim ∇IUn 0. Thus, we
prove the lemma.
4. Proof of Theorem 1.1
Assume that the conditions 1.5, 1.6, and 1.7 hold and F satisfies
F1, F2, F3, and F4.
We note that I0, 0 0. By iii of Lemma 2.4, U → grad Q Fx, t, u, vdx dt is a compact
mapping. By Lemma 3.2, there exists a small number ρ > 0 and a small ball Bρ ⊂ E with
radius ρ such that if U ∈ ∂Bρ , then α inf IU > 0, and there is an e ∈ E , R > ρ > 0 and a
large ball DR with radius R > 0 such that if
W DR ∩ E0 ⊕ E− ⊕ {re | 0 < r < R},
4.1
sup IU ≤ 0.
4.2
then
U∈∂W
Let us set β supW I. We note that β < ∞. Let En n be a sequence of subspaces of E
satisfying 3.2. Clearly E0 ⊂ En for all n, and ∂Bρ and ∂W link. We have, for all n ∈ N,
sup I < inf I.
∂W∩En
∂Bρ ∩En
4.3
Moreover, by Lemma 3.3, In I|En satisfies the P.S.∗c condition for any c ∈ R. Thus by
Lemma 3.1 linking theorem, there exists a critical point Un for In with
α ≤ inf I ≤ IUn ≤ sup I ≤ β.
∂Bρ ∩En
4.4
W∩En
Since In satisfies the P.S.∗c condition, we obtain that, up to a subsequence, Un → U, with
U a critical point for I such that α ≤ IU ≤ β. Hence, U /
0, 0. Thus, system 1.5 has a
nontrivial solution. Thus Theorem 1.1 is proved.
T. Jung and Q.-H. Choi
11
References
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external resonances,” Communications in Applied Analysis, vol. 3, no. 1, pp. 73–84, 1999.
3 Q.-H. Choi and T. Jung, “Multiplicity results for nonlinear wave equations with nonlinearities crossing
eigenvalues,” Hokkaido Mathematical Journal, vol. 24, no. 1, pp. 53–62, 1995.
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65 of CBMS Regional Conference Series in Mathematics, The American Mathematical Society, Providence,
RI, USA, 1986.